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Shekel function

<p>The Shekel function or also Shekel's foxholes is a multidimensional, multimodal, continuous, deterministic <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> commonly used as a test function for testing <a href="/facts/Optimization_(mathematics)/oRn8Iv5I">optimization</a> techniques.
</p><p>The mathematical form of a function in 
  
    
      
        n
      
    
    {\displaystyle n}
  
 dimensions with 
  
    
      
        m
      
    
    {\displaystyle m}
  
 maxima is:
</p><p>
  
    
      
        f
        (
        
          
            
              x
              →
            
          
        
        )
        =
        
          ∑
          
            i
            =
            1
          
          
            m
          
        
        
        
          
            (
            
              
                c
                
                  i
                
              
              +
              
                ∑
                
                  j
                  =
                  1
                
                
                  n
                
              
              (
              
                x
                
                  j
                
              
              −
              
                a
                
                  j
                  i
                
              
              
                )
                
                  2
                
              
            
            )
          
          
            −
            1
          
        
      
    
    {\displaystyle f({\vec {x}})=\sum _{i=1}^{m}\;\left(c_{i}+\sum \limits _{j=1}^{n}(x_{j}-a_{ji})^{2}\right)^{-1}}

</p><p>or, similarly,
</p><p>
  
    
      
        f
        (
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        .
        .
        .
        ,
        
          x
          
            n
            −
            1
          
        
        ,
        
          x
          
            n
          
        
        )
        =
        
          ∑
          
            i
            =
            1
          
          
            m
          
        
        
        
          
            (
            
              
                c
                
                  i
                
              
              +
              
                ∑
                
                  j
                  =
                  1
                
                
                  n
                
              
              (
              
                x
                
                  j
                
              
              −
              
                a
                
                  i
                  j
                
              
              
                )
                
                  2
                
              
            
            )
          
          
            −
            1
          
        
      
    
    {\displaystyle f(x_{1},x_{2},...,x_{n-1},x_{n})=\sum _{i=1}^{m}\;\left(c_{i}+\sum \limits _{j=1}^{n}(x_{j}-a_{ij})^{2}\right)^{-1}}

</p>